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Without using a calculator find the greatest prime factor of \(15^6 - 7^6\)

 Feb 16, 2018
 #1
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[7 + b]^6 - 7^6

b^6 + 42 b^5 + 735 b^4 + 6860 b^3 + 36015 b^2 + 100842 b + 117649 -7^6

Sub 8 for b

262,144 + 1,376,256 + 3,010,560 + 3,512,320 + 2,304,960 + 806,736

11,272,976 = 2^4 * 11 * 13^2 * 379

 Feb 16, 2018
edited by Guest  Feb 16, 2018
 #2
avatar+99580 
+1

15^6  - 7^6  =  {factor using  difference/sum of cubes}

 

(15 - 7)(15^2 + 15*7 + 7^2)(15 + 7) (15^2 - 15*7 + 49)  =

 

(15 - 7) (15+7)(15^2 - 15*7 + 49)(15^2 + 15*7 + 49) =

 

  (8)(22)(169)(379)

 

(2^3)(2*11)(13* 13)(379)

 

(2^4)(11)(13^2)(379)    is the largest prime factor

 

 

cool cool cool

 Feb 16, 2018
edited by CPhill  Feb 16, 2018
edited by CPhill  Feb 16, 2018
edited by CPhill  Feb 16, 2018
 #3
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A typo of 139 should be 169.

 Feb 16, 2018
 #4
avatar+99580 
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Thanks, guest.....correction made  !!!

 

 

cool cool cool

 Feb 16, 2018

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